Reciprocal relationships, reverse causality, and temporal ordering

Testing theories with cross-lagged panel models

Charles C. Lanfear

University of Cambridge

Thiago R. Oliveira

University of Manchester

Background

What is today about?

JDLCC paper; using neighb examples but paper will use life course

Reviewing common problems in applied work

Aligning theory, models, and estimation

Assume understand basics of conditional ignorability

Reciprocality

  • Theories sometimes imply reciprocality
  • Also sometimes an empirical problem
  • But pretty much nothing is actually reciprocal; the world is recursive
  • Accordingly, we use DAGs for theory, SEM diagrams for estimation

The Cross-Lagged Panel Model

X 1 X 2 X 3 Y 1 Y 2 Y 3

Simultaneous Effects

X 1 X 2 Y 1 Y 2 e x 2 e y 2

Mechanisms and time

  • Time enforces recursiveness
  • Things that appear non-recursive are usually omitting mediators or time

Show 5 time DAG, 3 time SEM

Do another with a mechanism

Time DAG

X 1 X 1 . 5 X 2 X 2 . 5 X 3 Y 1 Y 1 . 5 Y 2 Y 2 . 5 Y 3

Time Estimator

X 1 X 2 X 3 Y 1 Y 2 Y 3 e x 2 e y 2 e x 3 e y 3

Theory

  • Make your theory recursive; use a DAG
  • Estimation may requite non-recursive models

Three common problems

Temporal Order

  • Illustrated by Vaisey & Miles (2019)

  • If true model is \(y = \beta x_t + \alpha_i + e_{it}\) and you fit \(y = \beta^* x_{t-t} + \alpha_i + e_{it}\), \(E(\beta^*) = -0.5\beta\)

Example paper

Solutions

  • Hard: Use strong theory to get timing right
  • Use robust estimators, e.g., Allison#s approach
  • If ambiguous contemporaneous path matters, you’ll need strong instrument(s)
X 1 X 2 X 3 Y 1 Y 2 Y 3 e x 2 e y 2 e x 3 e y 3

Unobserved Heterogeneity

  • Most common concern is stable traits
  • Can’t just toss in fixed effects due to Nickell bias

Example: Lanfear & Kirk (2024)

A 1 U C 1 A 2 C 2 O 1 O 2 CE 1 CE 2

Collective efficacy (CE)
Crime (C)
Opportunity (O)

Airbnb properties (A)
Time-invariant unobservables (U)

Directionality assumptions render model recursive

Solutions

  • Different approaches
  • Psychology approaches: RI-CLPM
  • Econometric approaches: Arellano-Bond, ML-SEM
    • Also relax strict exogeneity

Low Temporal Variation

  • \(Var(Y_2|Y_1) \rightarrow 0\) as \(\rho(Y_1,Y_2) \rightarrow 1\)
  • Error becomes proportionally larger component
  • Common with narrow waves, stable constructs

Example paper

That neighbs one? Hipp?

Solutions

  • Error part can be dealt with using measurement models
    • Should do this anyway as outcomes are regressors, so measurement error attenuates estimates
  • Address during data collection
    • Oversample for change
    • Look for shocks
  • Consider different time lags or units of analysis
  • Might be nothing you can do

Advice

  • Theory comes first
  • Default to robust methods

Feedback and Questions

Contact:

Charles C. Lanfear
Institute of Criminology
University of Cambridge
cl948@cam.ac.uk